 LAPACK 3.3.1 Linear Algebra PACKage

# ssysvx.f

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```00001       SUBROUTINE SSYSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B,
00002      \$                   LDB, X, LDX, RCOND, FERR, BERR, WORK, LWORK,
00003      \$                   IWORK, INFO )
00004 *
00005 *  -- LAPACK driver routine (version 3.3.1) --
00006 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00007 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00008 *  -- April 2011                                                      --
00009 *
00010 *     .. Scalar Arguments ..
00011       CHARACTER          FACT, UPLO
00012       INTEGER            INFO, LDA, LDAF, LDB, LDX, LWORK, N, NRHS
00013       REAL               RCOND
00014 *     ..
00015 *     .. Array Arguments ..
00016       INTEGER            IPIV( * ), IWORK( * )
00017       REAL               A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
00018      \$                   BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
00019 *     ..
00020 *
00021 *  Purpose
00022 *  =======
00023 *
00024 *  SSYSVX uses the diagonal pivoting factorization to compute the
00025 *  solution to a real system of linear equations A * X = B,
00026 *  where A is an N-by-N symmetric matrix and X and B are N-by-NRHS
00027 *  matrices.
00028 *
00029 *  Error bounds on the solution and a condition estimate are also
00030 *  provided.
00031 *
00032 *  Description
00033 *  ===========
00034 *
00035 *  The following steps are performed:
00036 *
00037 *  1. If FACT = 'N', the diagonal pivoting method is used to factor A.
00038 *     The form of the factorization is
00039 *        A = U * D * U**T,  if UPLO = 'U', or
00040 *        A = L * D * L**T,  if UPLO = 'L',
00041 *     where U (or L) is a product of permutation and unit upper (lower)
00042 *     triangular matrices, and D is symmetric and block diagonal with
00043 *     1-by-1 and 2-by-2 diagonal blocks.
00044 *
00045 *  2. If some D(i,i)=0, so that D is exactly singular, then the routine
00046 *     returns with INFO = i. Otherwise, the factored form of A is used
00047 *     to estimate the condition number of the matrix A.  If the
00048 *     reciprocal of the condition number is less than machine precision,
00049 *     INFO = N+1 is returned as a warning, but the routine still goes on
00050 *     to solve for X and compute error bounds as described below.
00051 *
00052 *  3. The system of equations is solved for X using the factored form
00053 *     of A.
00054 *
00055 *  4. Iterative refinement is applied to improve the computed solution
00056 *     matrix and calculate error bounds and backward error estimates
00057 *     for it.
00058 *
00059 *  Arguments
00060 *  =========
00061 *
00062 *  FACT    (input) CHARACTER*1
00063 *          Specifies whether or not the factored form of A has been
00064 *          supplied on entry.
00065 *          = 'F':  On entry, AF and IPIV contain the factored form of
00066 *                  A.  AF and IPIV will not be modified.
00067 *          = 'N':  The matrix A will be copied to AF and factored.
00068 *
00069 *  UPLO    (input) CHARACTER*1
00070 *          = 'U':  Upper triangle of A is stored;
00071 *          = 'L':  Lower triangle of A is stored.
00072 *
00073 *  N       (input) INTEGER
00074 *          The number of linear equations, i.e., the order of the
00075 *          matrix A.  N >= 0.
00076 *
00077 *  NRHS    (input) INTEGER
00078 *          The number of right hand sides, i.e., the number of columns
00079 *          of the matrices B and X.  NRHS >= 0.
00080 *
00081 *  A       (input) REAL array, dimension (LDA,N)
00082 *          The symmetric matrix A.  If UPLO = 'U', the leading N-by-N
00083 *          upper triangular part of A contains the upper triangular part
00084 *          of the matrix A, and the strictly lower triangular part of A
00085 *          is not referenced.  If UPLO = 'L', the leading N-by-N lower
00086 *          triangular part of A contains the lower triangular part of
00087 *          the matrix A, and the strictly upper triangular part of A is
00088 *          not referenced.
00089 *
00090 *  LDA     (input) INTEGER
00091 *          The leading dimension of the array A.  LDA >= max(1,N).
00092 *
00093 *  AF      (input or output) REAL array, dimension (LDAF,N)
00094 *          If FACT = 'F', then AF is an input argument and on entry
00095 *          contains the block diagonal matrix D and the multipliers used
00096 *          to obtain the factor U or L from the factorization
00097 *          A = U*D*U**T or A = L*D*L**T as computed by SSYTRF.
00098 *
00099 *          If FACT = 'N', then AF is an output argument and on exit
00100 *          returns the block diagonal matrix D and the multipliers used
00101 *          to obtain the factor U or L from the factorization
00102 *          A = U*D*U**T or A = L*D*L**T.
00103 *
00104 *  LDAF    (input) INTEGER
00105 *          The leading dimension of the array AF.  LDAF >= max(1,N).
00106 *
00107 *  IPIV    (input or output) INTEGER array, dimension (N)
00108 *          If FACT = 'F', then IPIV is an input argument and on entry
00109 *          contains details of the interchanges and the block structure
00110 *          of D, as determined by SSYTRF.
00111 *          If IPIV(k) > 0, then rows and columns k and IPIV(k) were
00112 *          interchanged and D(k,k) is a 1-by-1 diagonal block.
00113 *          If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
00114 *          columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
00115 *          is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) =
00116 *          IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
00117 *          interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
00118 *
00119 *          If FACT = 'N', then IPIV is an output argument and on exit
00120 *          contains details of the interchanges and the block structure
00121 *          of D, as determined by SSYTRF.
00122 *
00123 *  B       (input) REAL array, dimension (LDB,NRHS)
00124 *          The N-by-NRHS right hand side matrix B.
00125 *
00126 *  LDB     (input) INTEGER
00127 *          The leading dimension of the array B.  LDB >= max(1,N).
00128 *
00129 *  X       (output) REAL array, dimension (LDX,NRHS)
00130 *          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
00131 *
00132 *  LDX     (input) INTEGER
00133 *          The leading dimension of the array X.  LDX >= max(1,N).
00134 *
00135 *  RCOND   (output) REAL
00136 *          The estimate of the reciprocal condition number of the matrix
00137 *          A.  If RCOND is less than the machine precision (in
00138 *          particular, if RCOND = 0), the matrix is singular to working
00139 *          precision.  This condition is indicated by a return code of
00140 *          INFO > 0.
00141 *
00142 *  FERR    (output) REAL array, dimension (NRHS)
00143 *          The estimated forward error bound for each solution vector
00144 *          X(j) (the j-th column of the solution matrix X).
00145 *          If XTRUE is the true solution corresponding to X(j), FERR(j)
00146 *          is an estimated upper bound for the magnitude of the largest
00147 *          element in (X(j) - XTRUE) divided by the magnitude of the
00148 *          largest element in X(j).  The estimate is as reliable as
00149 *          the estimate for RCOND, and is almost always a slight
00150 *          overestimate of the true error.
00151 *
00152 *  BERR    (output) REAL array, dimension (NRHS)
00153 *          The componentwise relative backward error of each solution
00154 *          vector X(j) (i.e., the smallest relative change in
00155 *          any element of A or B that makes X(j) an exact solution).
00156 *
00157 *  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
00158 *          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
00159 *
00160 *  LWORK   (input) INTEGER
00161 *          The length of WORK.  LWORK >= max(1,3*N), and for best
00162 *          performance, when FACT = 'N', LWORK >= max(1,3*N,N*NB), where
00163 *          NB is the optimal blocksize for SSYTRF.
00164 *
00165 *          If LWORK = -1, then a workspace query is assumed; the routine
00166 *          only calculates the optimal size of the WORK array, returns
00167 *          this value as the first entry of the WORK array, and no error
00168 *          message related to LWORK is issued by XERBLA.
00169 *
00170 *  IWORK   (workspace) INTEGER array, dimension (N)
00171 *
00172 *  INFO    (output) INTEGER
00173 *          = 0: successful exit
00174 *          < 0: if INFO = -i, the i-th argument had an illegal value
00175 *          > 0: if INFO = i, and i is
00176 *                <= N:  D(i,i) is exactly zero.  The factorization
00177 *                       has been completed but the factor D is exactly
00178 *                       singular, so the solution and error bounds could
00179 *                       not be computed. RCOND = 0 is returned.
00180 *                = N+1: D is nonsingular, but RCOND is less than machine
00181 *                       precision, meaning that the matrix is singular
00182 *                       to working precision.  Nevertheless, the
00183 *                       solution and error bounds are computed because
00184 *                       there are a number of situations where the
00185 *                       computed solution can be more accurate than the
00186 *                       value of RCOND would suggest.
00187 *
00188 *  =====================================================================
00189 *
00190 *     .. Parameters ..
00191       REAL               ZERO
00192       PARAMETER          ( ZERO = 0.0E+0 )
00193 *     ..
00194 *     .. Local Scalars ..
00195       LOGICAL            LQUERY, NOFACT
00196       INTEGER            LWKOPT, NB
00197       REAL               ANORM
00198 *     ..
00199 *     .. External Functions ..
00200       LOGICAL            LSAME
00201       INTEGER            ILAENV
00202       REAL               SLAMCH, SLANSY
00203       EXTERNAL           ILAENV, LSAME, SLAMCH, SLANSY
00204 *     ..
00205 *     .. External Subroutines ..
00206       EXTERNAL           SLACPY, SSYCON, SSYRFS, SSYTRF, SSYTRS, XERBLA
00207 *     ..
00208 *     .. Intrinsic Functions ..
00209       INTRINSIC          MAX
00210 *     ..
00211 *     .. Executable Statements ..
00212 *
00213 *     Test the input parameters.
00214 *
00215       INFO = 0
00216       NOFACT = LSAME( FACT, 'N' )
00217       LQUERY = ( LWORK.EQ.-1 )
00218       IF( .NOT.NOFACT .AND. .NOT.LSAME( FACT, 'F' ) ) THEN
00219          INFO = -1
00220       ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) )
00221      \$          THEN
00222          INFO = -2
00223       ELSE IF( N.LT.0 ) THEN
00224          INFO = -3
00225       ELSE IF( NRHS.LT.0 ) THEN
00226          INFO = -4
00227       ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
00228          INFO = -6
00229       ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
00230          INFO = -8
00231       ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
00232          INFO = -11
00233       ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
00234          INFO = -13
00235       ELSE IF( LWORK.LT.MAX( 1, 3*N ) .AND. .NOT.LQUERY ) THEN
00236          INFO = -18
00237       END IF
00238 *
00239       IF( INFO.EQ.0 ) THEN
00240          LWKOPT = MAX( 1, 3*N )
00241          IF( NOFACT ) THEN
00242             NB = ILAENV( 1, 'SSYTRF', UPLO, N, -1, -1, -1 )
00243             LWKOPT = MAX( LWKOPT, N*NB )
00244          END IF
00245          WORK( 1 ) = LWKOPT
00246       END IF
00247 *
00248       IF( INFO.NE.0 ) THEN
00249          CALL XERBLA( 'SSYSVX', -INFO )
00250          RETURN
00251       ELSE IF( LQUERY ) THEN
00252          RETURN
00253       END IF
00254 *
00255       IF( NOFACT ) THEN
00256 *
00257 *        Compute the factorization A = U*D*U**T or A = L*D*L**T.
00258 *
00259          CALL SLACPY( UPLO, N, N, A, LDA, AF, LDAF )
00260          CALL SSYTRF( UPLO, N, AF, LDAF, IPIV, WORK, LWORK, INFO )
00261 *
00262 *        Return if INFO is non-zero.
00263 *
00264          IF( INFO.GT.0 )THEN
00265             RCOND = ZERO
00266             RETURN
00267          END IF
00268       END IF
00269 *
00270 *     Compute the norm of the matrix A.
00271 *
00272       ANORM = SLANSY( 'I', UPLO, N, A, LDA, WORK )
00273 *
00274 *     Compute the reciprocal of the condition number of A.
00275 *
00276       CALL SSYCON( UPLO, N, AF, LDAF, IPIV, ANORM, RCOND, WORK, IWORK,
00277      \$             INFO )
00278 *
00279 *     Compute the solution vectors X.
00280 *
00281       CALL SLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
00282       CALL SSYTRS( UPLO, N, NRHS, AF, LDAF, IPIV, X, LDX, INFO )
00283 *
00284 *     Use iterative refinement to improve the computed solutions and
00285 *     compute error bounds and backward error estimates for them.
00286 *
00287       CALL SSYRFS( UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X,
00288      \$             LDX, FERR, BERR, WORK, IWORK, INFO )
00289 *
00290 *     Set INFO = N+1 if the matrix is singular to working precision.
00291 *
00292       IF( RCOND.LT.SLAMCH( 'Epsilon' ) )
00293      \$   INFO = N + 1
00294 *
00295       WORK( 1 ) = LWKOPT
00296 *
00297       RETURN
00298 *
00299 *     End of SSYSVX
00300 *
00301       END
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